Wave transmission network



A ril 26, 1938. s. DARLINGTON WAVE TRANSMISSION NETWORK 2 Sheets-Sheet 1 Filed March 20, 1955 FIG.

LINE /5 HIGH-PASS LOW-PASS LOW-PASS HIGH-PASS lNl/ENTOR' .S. DARL/NGTON ATTORNEY April 26, 1933. s DARUNGTON 2,115,138

WAVE TRANSMISSION NETWORK Filed March 20, 1935 2 Sheets-Sheet 2 FIG. 7

IN l/ENTOR 5. 0A RL/NG TON Mw K A TTORNEY Patented Apr. 26, 1938 UNITED STATES PATENT OFFICE A WAVE TRANSMISSION NETWORK Sidney Darlington, New York, N. Y., assignor to Bell Telephone Laboratories, Incorporated, New York, N. Y., a corporation of New York Application March 20, 1935, Serial No. 11,934

14 Claims.

This invention relates to wave transmission networks and, more particularly, to networks having frequency selective transmission characteristics.

Frequency selective transmission networks having the property of constant resistance characteristic impedance have been known and used for some time but heretofore only in the form of two-terminal impedances or of ordinary fourterminal networks. An object of the present invention is to extend this constant resistance property to more complex structures, in particular to eight terminal networks. Networks of this type comprise four individual transmission paths and are useful for the separation of currents of diiferent frequencies as, for example, in carrier current repeaters. The problem of connecting thesenetworks into a given transmission system is greatly simplified by the present invention since, by virtue of the constant resistance characteristic, reflection effects at the insertion points are greatly reduced and, in many cases, substantially eliminated.

The nature and the underlying principles of the invention will be more fully understood from the following detailed description and by refer- 7 system in accordance with the invention.

The network shown in Fig. 1 comprises two parallel transmission paths extending between pairs of terminals I, 2 and 3, 4, and each including two component networks 9, ID, in the upper path, and II, I2, in the lower path. Resistances of value R are connected between terminals I, 2 and 3, 4, and also between two additional pairs of'terminals 5, 6 and l, 8, located between networks 9 and I0 and networks II and I2, respectively.

The networks 9, II], iI and I2 are of the symmetrical lattice type and are similar in pairs, networks 9 and I I being alike and also networks III and I2. The line and the lattice branch impedances of networks 9 and I I have values A and B, respectively, and the corresponding branch impedances of networks It and 12 have values C and D, respectively. These impedances are preferably pure reactances and may be chosen to give each network a desired type of transmission characteristic. For example 9 and II may have low-pass characteristics while II] and I2 may be of the'high-pass type.

The order of the component networks is reversed in the one path with respect to the other, but otherwise the two paths have the same composition and therefore have equal transfer constants. There is also a reversal in the intercon-' nection of the two paths at terminals 3 and 4 with respect to the connections at I and 2, the effect of which is equivalent to the addition of a phase shift of 180 degrees in one or other of the two I paths. I

This phase reversal, together with the similarity between the two paths gives rise to a condition of conjugacy between the resistance paths at terminals I, 2 and 3, l, respectively. This may be demonstrated as follows: Consider the resistance R between terminals 3 and 4 to be replaced by a short circuit and a Voltage Eu applied to, terminals l and 2. Under this condition the total current through the short circuit will be the resultant of the separate transmissions through the individual paths, the output from the one path being unable to pass beyond the short circuit into the other path. From the principle of reciprocity it follows that the current in the short circuit from either one of the paths alone will not be changed if the path is turned end for end. Since such a reversal in one path would make the two paths exactly alike, it follows that the output,

currents in the short circuit will be of equal magnitude and will completely neutralize each other because of the relative reversal of phase due to the circuit connections. Since no current flows in the short circuit an impedance of any magnitude may be inserted therein without disturbing the conjugacy. Similar considerations will show that the resistance paths between terminals 5 and 6 and between terminals 1 and 8 are also cone jugate. In this case the two paths include net- 'works'lfl, II and 9, I2 respectively, with intermediate bridging resistances, and are similar in composition to the two paths considered in the previous case. A phase reversal is also present in one path with respect to the other due to the interconnections at the intermediate terminals I, 2 and 3, I.

When the four pairs of terminals are bridged by equal resistances of value R it is possible, by proper choice of the impedances of the component networks, to make the input impedances of the whole network at each of the four pairs of terminals equal to the resistance R at all frequencies. This is the same thing as giving the system a constant resistance image impedance of value R at all four of its pairs of terminals, the image impedances of a transmission network be ing defined as the impedances measured at each of the several pairs of terminals when the other pairs image impedances.

are closed through impedances which produce zeroreflection eflects at all fre'quencies that iswhen" they are closedthrough their. respective acte'ristic may be found as"fol1ows: ,w Consider the'input impedance at terminals 1 V and 2. Since the resistance path'b etween 3 and 4 is conjugate to the path betweenv l, and g; the input impedance will not be. affected by the value of the resistance in the branch 3, 4. "The termi nals 3 and 4 may therefore be assumed to' be of the r-other transmission path a'lso short cir-f Qcuite'd at the output e'nd, then; if Z denotethe; '25 a fshort-circuited in'which case.the input impedahce at! and '2 is simply the" impedance of the I a 1 two transmission paths in parallel Let Z1 denote the impedance of the" path including networks 9, and II] was the joutputpterminals of I and let Z2 denote, the impedance snort-dreamed total input impedance: at land 2,

1'11 71 19 .'The values of Zr and Z2 may be determined from theope'n circuit and short circuit impedances :ofv the component networks. "Since each of these networks isfsymmetrical' its open and short circuit'impedances will be the same ;at; both ends.

- Networks Sand H', being similar, will have the same impedances'and' networks ID and -I 2 likewise. Let 'the'open'circuit andshort circuit 'imf 7 "sedans seer 9 and l I" be denoted Z0 and Zs re sp'ectivel'y' and let; the corresponding impedances V of networks land 12 be denotedtbyYo and Y5.

"'The'im'pedance Z1 is that ofthe network 9 terminatedby.an impedance made up 'of the parallel combination of resistance R and-the short value circuit impedance of network I0 and has the r1+ a V. Z=Z l .1+Z.. W2

'Where Zrr m Substituting 7 the; value of Z71, i Equation '2' and inverting, gives titre-(mal V i 2; (Y.+Z.)+Y.Z.' In a like manner the value ofZz is found to be 1 1 RZ +Y R+Z r r (a) If the inputimpedance at terminals l, 2 is to be a constant resistance of value R then Z1 and'Zz must be so related that V 7 from which, by simplification, the relationship is Hobtained. r j V 1 1 .2 1 F Z 'ZfY f'I/ Y terms of the line and lattice impedancesp 7 ..Infsymmetrical systems the V V imageimpedances are equala'nd are equal to the characteristic impedance. Thednecessary 111619:- tionships between the Vcomponent'network' im-i' pedances to achieve this constant resistancechar- Vmiss'on lines lattices 0 A, B, C and D, which make up'the networks 1 Sto l2, Equation 6 becomes 1 .1 '1 i i: 11 r v "-e-A"B eTJ E(K+B (C D 7 Equations 6 and 7 set forth relationships or the network impedances, and of the branch impedwhich may be followed in choosing the impedances A, B, C'and' D in accordance with these requirements will be described later. If will be sufficient atthis point to note that the requireinents can always be met with physical imped- 1 ance elements and that these inay be selected andproportioned 'to provide desired transmission between adjacent pairs of ter-v characteristics n le v a 1 In the network of Figure 1 the two transmission paths are 'ofjthe balanced type and the networkis therefore suitedIfor direct connection to transground. "A modified form of the invention in which the two transmission rams are of the unbalanced type 'is shown Figure 2.] In this network the component'net'works are desig'nated i 9',- 59, ll, and IZf and'consist offseries-shunt impedance combinations with the series Vimpedances inserted in only one line of each path.

' A procedure "ances when'lattice type'networks are used, which must be fulfilled in order that the constant resistance requirement" may be met. 10

having "both sides balanced 130 Terminals 2, 4, -6 and Bare connected together and'to ground. 7 o permit the grounding of oneside of eachfpath a transformerTrisincluded at one end ofthe upper. path was second trans:

former T2 at the other. end of the lower path.

These transformers should besixnilar in their characteristics and should have unity transformation ratios. If theyare similarly poled the relative phase shifts of the two paths will not be afiected by their inclusion and the conjugacy ofthe opposite pairs of terminalswill not be impaired. The component networks in Figure 2 are shown as symmetrical T networks, .8 and- II haVing series impedances F and shunt im-V pedances G and I6 and 12' having corresponding'fimpedances H and K. Single sec'tionnet-I works only are shown, but it will be understood that as mans sections as desired may be used;

the showing being intendedto-representsimply the unbalanced equivalents of zthe generalized Figure 1. 'It should'be noted, however, that the component networks are of the series terminated type. 'Thisis necessary where the resistances'B, are connected in shunt to the component networkterminals; as in Figures l and 2.

,Otherwisaia shunt path would be. provided at each pair of terminals-which would have zero V impedance at some frequency and wouldthr'e- I fore prevent the realizationof the constant resistance characteristic.

. Another form of the inventionis shown in Figure 3 which differs from that shown in Fig instead of in shunt.

In the latter case the lower line of the upper path and the upper line of the lower path would form the common grounded conductor. Transformers Ti and T2 inserted as in Figure 2 permit 'the'grounding to be effected and, with appropriate poling, maintain the conjugacy of the opposite pairs of terminals. When the unbalanced networks are used they should be of the shunt terminated type for reasons converse to those dictating the use of series terminations in the network of Figure 2.

From considerations similar to those applied to the analysis of the circuit of Figure 1 it may be shown that the relationship required in this case for securing a constant resistance impedance characteristic at each of the four pairs of terminals is expressed by o s+ c+ o a or, in terms of the lattice impedances A, B, C and D (C+ CD An example of the use of the invention is illustrated in Figure 4 which shows its application to a carrier telephone or telegraph repeater of the single amplifier type shown in U. S. Pat. 1,874,492 issued Aug. 30, 1932, to A. G. Ganz. The network illustrated corresponds to that of Figure l in which the resistances are bridged across the terminals of the component networks. The two portions I4 and 15 of a transmission line in which the repeater is inserted are connected to terminals l, 2 and 3, respectively and furnish the requisite bridging resistances. An amplifier i3 is connected between terminals 5, 6 and l, 8. The bridging resistances being provided by suitable elements associated with the input and output circuits of the amplifier.

In a multiplex carrier transmission system it is customary to include all channels transmitting in one direction in a low frequency group and those transmitting in the opposite direction in a separate high frequency group. By making networks 9 and i i of the character of low-pass filters a path through the rep-eater from line I4 to line [5 is provided for the low frequency channels, this path including filter 9, amplifier l3 and filter II. By making networks in and I2 of the character of high pass filters a corresponding path through the repeater from line l5 to line I 4 is provided for the .high frequency channels. These paths may be made mutually exclusive by the proper choice ofthe filter cut-off frequencies and by providing adequate attenuation in the filters.

The conjugacy existing between the opposite pairs of terminals because of the reversal of the interconnections at terminals 3 and 4 with respect to those at terminals l and 2 not only permits the constant resistance characteristic to be obtained but also minimizes the possibility of singing in the amplifier by the reduction or elimination of feed-back from the amplifier output terminals '3, 8 to the input terminals 5, 6.

The general requirements for the constant resistance condition of the networks have been set forth in Equations 6, '7, 8 and 9. It remains to be shown how the individual impedances of the networks may be determined in accordance with these requirements. This will be done by developing design formulae for a network of the type of Figure 1, the general procedure outline being applicable to the other forms of network.

Equation 7 which expresses the requirement for the constant resistance condition in terms of the branch impedances is general to the extent that it does not depend on the character of the impedances, which may be resistive or reactive or may include both resistance and reactance. However, since it is desirable in practice'that the sys-.

tem should have definite selective properties the component networks will preferably be composed of substantially pure reactances and will have individual selective characteristics corresponding to the prescribed requirements of the system. With this in mind the illustrative example will correspond to the arrangement shown in Figure 4 in which networks 9 and II have low-pass transmission characteristics and networks 49 and I2 have high-pass characteristics.

Assuming the lattice branches to be substantially pure reactances and writing in place of the impedances A, B, C and D the corresponding reactances X5, Xb, Xc, and Xd, Equation 7 can be transformed to 1 1 1 1 2 4 a '(10) or 1 1 1 1 2 4 1 2 =(z+ F 5 (11) where and and that between terminals I, 2 and 1, 8 by where a1 and a2 are the respective insertion losses.

Since it is desired that the path through network 9 have a low-pass characteristic and that through network I2 a high-pass characteristic, it is apparent from Equations 12 and 13 that the ratio FZ/Fl must be small in the low-pass range and large in the high-pass range. That is, F2 must be small in the range where F1 is large and vice versa.

The character of the frequency variations of F1 and F2 may readily be determined in any par ticular case. A suitable form for the low-pass networks 9 and l I is illustrated in Figure 5 in which X3, is characterized by a single finite resonance and Xb by two finite resonances. From considerations of ordinary filter theory the resonance of X1; will occur at a frequency lying between the two resonance frequencies of Xe.

In terms of the resonance frequencies F1 may be expressed by A20) A460 A660 where w denotes 2 11' times frequency, (012, (022, and 0132 correspond to the three resonance frequencies, and A0, A2, A4 and A6 are constants. The resonance frequencies all occur in the low-pass transmission range and at these frequencies F1 become infinite. v 7

. Equation 14 may rbe transformed byordinary' algebraic processes to the. form V where and win; am, am, correspond to frequencies at which F1 has zero values.

7 These frequencies, if real, all lie above the low-' pass range; but in the general case certain of i them'may be imaginaryor complex.

F1 in the factorial The general expression for r 2d form of Equation 15 ay be written as i of zeros.

- Bypa proper choicerof critical frequencies dee fining the zeros and poles in- Equations 15 and 16 f the'rfrequency Va 11atiOn OfIF1g may. be made to -take the form illustrated by the curve inflFig. 6

which is characterized by a seriesof equal mine ima'betwe'en the poles in the low frequency range and a corresponding'series' of equal maxima be .tween the Zeros in the'flhig'h frequency irange.

Moreover the. values of F1 in the high frequency 7' range may be made so small as to be completely 7 negligible. 7

. frequencies and The design proceeds by determining thecriti cal the element values of the lowpass network so thatthe minima of F? in the low frequency range are equal to 4/R and so that the descending portion of the F1 characteristic in the range through the value 2/R at a predetermined frequency marking the division of the ranges.

'A complementary characteristic is next deterimined for 'Fz having minima in the highfrequency range likewise equalto 4/3 andhaving its descending portion "intersect that of F1 at the predetermined dividing frequency, itsslo-pe at V this point being equal that of F1 but of opposite sign. This ensures a minimum at' the dividing" frequency the value of which will be i R Since the "values of F1 in the high. frequency 7 7 range are negligible and those of Fg Jin the low frequency range are likewise negligiblethe sum of .the two quantities gives. a'series' o f minima' eacheof value 4/13? and alternating in the fre quency'scale with'a series' of infinite. values as It' develops also that V the form ofVF1 +F2 agrees with that of the right required by Equation 11;

7 hand side of Equation ll and hence that the coristant resistance requirement is met. I have found that'the neglect of. the low values of F1 in the 7 high frequency range'and of theecorresponding values of F2 affects the resistance of the system to an extent of less than one part in 5000 inlpractical i n a Ihe choiceof the critical frequencies of F1? to I give therequired frequency variation is based on p indicated quantity goes through a series of equal cases.

7 the following expansion theorems for elliptic functions which follow from relationships given in Elliptic Functions, by Cayley, published by.

betweenfthe high and the low values passes 1 tionship tity -G. 'Bell and Sons, London, England,

tion, 1895, pages 265 and 267.

Esn(uK, k)

wherein the following notations are used:

V Expressions of the form sn (y,.a)'denote elliptic;-

' sines of modulus a, andargument y, the modulus 7 being apositive numeric of value less than unity;

K is the complete elliptic integral or the first kind.

of modulus k; j

r s the. complete elliptic integral 'of the first; kind,'ofmodu1usk;; J

' nf--'21 r i +1 sn(uK,k). is the variable parameter.

following manner.

Let K and :K1 denote the complete. elliptic. 1h

tegrals'ofthefirst kind, of'moduli V Knowing q and n the value of'q1 is determined second edi- C2, C3 and C4 are numerical constants and j J:

r The q a t s K, hk, k are reeteem the r '1 f .I' "it and the corresponding value of 701 can be found.

from standard tables of elliptic functions. 7 For the cases of interest in connection'with thepres 7 ent invention k1 is small and maybe determined with great accuracy expressed as afunctioh ofjsnill; K, 70) has poles, orinfinite valuescorresponding to the z ero .values .of the .denominator factors; and has an equal number of related zeros which occur in a different range" of values 'of the argument sn l (u, K, k). In accordance with'the knownprin'ci r the, flpp f mate rela- Ye l I ples 'of elliptic functions the'square'of the above minima between the poles maximabetween the zeros.

The values of the minima are given by Q1} kl V (24) and the maxima by 1 1 the ratio of the maxima to the minima being equal to 7612.

The constants C3 and C4 are found by comparing the expressions in which they appear as Sn (H K, k) approaches infinity. This gives rise to the relationship Since the variation of the function and the variable frequency ratio wo/w is identified with where The value of the modulus 7c is chosen with reference to the degree of discrimination required of the network. As the value of the modulus approaches unity the poles and zeros of F1 move close to we and the characteristic is marked by low minima and high maxima. Reducing the value of the modulus separates the critical frequencies and at the same time increases the ratio of the minima to the maxima. In the case of a network of small degree of complexity, for example one having two poles and two zeros, a modulus value of 0.9 will ensure a satisfactorily sharp discrimination by the network and will result in a ratio of the minima to the maxima greater than 5000. For more complex networks higher values of the modulus may be taken and because of the larger number of poles and zeros both the sharpness of discrimination and the ratio of the minima to the maxima are greatly increased.

In the case of the network illustrated in Fig. 5, for which the function F1 has three poles and three zeros, Equations 26 and 27 give the following values for the critical frequencies 4031 (no-Z- ICSH(%K, k) Log/p3 22 o 11 and a1 a u Substituting these values in the right hand side of Equation 18 and identifying this with F1 gives which agrees with Equation 15 when Substitution of the same values in the right hand side of Equation 1'7 gives This equation corresponds to Equation 14 and enables the constants A0, A2, etc. in that equation to be determined in terms of the chosen elliptic function parameters so that the desired variation of F1 will be secured. Since each term in Equation 14 represents the susceptance of a branch path containing only a simple inductance or the combination of an inductance and capacity in series the determinations of the As leads directly to the element values.

The required value of C3 follows from Equations 25 and 26. Equation 25 gives the value of the 'minima of F1 between the poles, which, in accordance with the design requirements must be equal to 4-:-R Accordingly and hence, from Equation 26 2 /EK R 111( (31) For the cases of interest, that is where the networks have relatively sharp discrimination and high attenuation, the value of K1 will not differ sensibly from 11/2 and the approximation for C3 C 4 /21: 3 'R 1'11:- maybe used.

The high pass network, for which the function F2 has to be complementary to F1 has to meet the requirement that its function F2 must have negligibly small values in the low pass range, have minima equal to 4+R in the high pass range, and must have the rising part of its characteristic intersect the descending part of F1 at the point 2+R with a slope equal to that of F1 and opposite in sign.

These requirements are most readily met by making the schematic form of the network such that F2 has the same number of poles and zeros as F1 and locating the poles and zeros symmetrically with the corresponding poles and zeros of F1 with respect to the cross over frequency, which will be denoted by wc+21r. The poles of F1 and F2 will thus occur in pairs having we as their geometric mean and the zeros will likewise occur in similarly related pairs.

The cross over point we will not be the same as we which appears in the formulae for F1 but will be slightly lower. The mathematical determina: tion of this point is rather lengthy and only the 7 therefore be derived from Equation 29 by the sim- T explicit formula for the ratio of t6 t9 at willibe In design practice the valuewc will generally be assigned. Equation 33 then permits the value am for the function F1 to be determined and also,

from the required symmetry of the two systems,

a corresponding value which will be denoted 101i 7 Qq for the function F2. {The values of m0 and too are related to we by the equation Using the same principles 7 as discussed above' formulae for F;v corresponding to Equations 28 and 29 can'be. developed,-but it'is simpler'to determine F2 directly from Equation 29 taking ad 6 vantage of the symmetry of the Fi and-F's char 7 wacteristics, For each value of m above we the func- 'tion Fz will have the same value as Fi at the pleexpedient of replacing w therein by "oc +w. The. negativesign appears in thistransformation .to take accountof'the difference in-rsign of the reactancesof coilsand of condensers whichjrepresent complementaryimpedances. l

Thezte'rms of the expression thus willi'ncliide two kinds; one set will directly represent physically realizable V susceptances and the remainder will represent physically realizable susceptances when the signjis changed. ,Those representing directly realizable susceptances are idBHtifiEd With Xc andthe others with Xe. The

schematic form of the high-pass network thus obtained isillustrated in Fig. '7. The reactance Xe comprises a capacity and a resonant circuit connected in parallel and the 'rea'ctance Xd' com- I prises two simple resonant circuitsconnectedin I 1 parallel.

Fig; 8 showsthe curves" of the two functions F1 and F2 and their sum plotted against the loga rithm of frequency. Curve h'l represents F1 curve H-representsFs and the looped portions oi thetwocurves together with dotted curve I2 represents their sum; The crossover frequency is designated by f0, the poles of F1? by'f12,j22,

= and fsz, and the poles of F2 by the inversely re- .of networkushown' in Fig. 2.

lated frequencies 'f'iz, f zz and $32.. .The sym- Imetry otthetwo characteristics about fc is clear- 1y shown by: this figure.

Whilejthe foregoing procedures give the de-' sired networks as symmetrical lattices, these may be'reduced by known procedures to unbalanced types of networks such as bridged-r-T ;or ladder networks suitable for use in the unbalanced type What is claimed is: e

1. i A; wave transmission network comprising two.

pathslconnected in parallel betweena, first pair 6'5 o f termin'als and a second pairof terminals, a "having; complementary transmission bands con:

pair of symmetrical frequency selectivenetworks nected; in tandemin one of said paths, apair of respectively :similar networks connected in ta n,

; am 1 he,(thet..of s id; pathsgbut in reverse orderwithlrespect to the; networks in saidifirst path; andmeans for reversing the phase of the currents in one of, said paths with'res pect to-the currents: the other whereby said pairs of ter'-. minals are made to be conjugate to each other 2. A wave transmissionnetwork comprising two with the equation found in. F2 7 paths connectedin parallel between a first pair of terminals and a second pair of terminals, a pair of symmetrical frequency selective networks having complementary transmisison bands. 'connected in tandem in one of said paths, apair of respectively similar networksconnected in tan- "dem in the other of said paths, butin reverse order with respect to thenetworks in the said first paths, 'saidnetworks having open circuit and short circuit 'impedances related in accordance Where Z0 and Z5 are respectively the open circuit andthe short circuit impedances of the one pair of similar networks, Y0 and Y5 are the cortransmission network -comprising 3. A wave;

two paths connected in parallelbetween a first pair of terminalsand a second pair of terminals, a a pair of symmetrical frequency selective networks having complementary transmission bands connected in tandem in one of said paths, a pair of respondingvalues for the other pair of similar networks, and R is an arbitrarily assigned res-istance.

respectively similar networks connected in tandem in the other of saidpaths, but in reverse order with respecttothe networks in the said first paths, said networks having open circuit and short circuit impedances relatedin accordance with the equation fR2=ZoZs+ 2YoYs+YoYs where Z; andZg are respectively the open'circuit and the short circuit impedancesof the one pair 4 A transmission network comprising two tially complementary. transmission bands diS.- posedin tandem in one of said paths, and two 'respectivelysimilar networks disposed in tandem in reverse, order-in the other oi said paths, the networks off one of the similar pairs being defined by two frequency characteristics 11( 0) and not which-jointly determine'their transmission prop;

cities, and the networks of p the, other similar pair; J beinglliklewise defined by frequency characteristics Js(w) and ,f4(w),and the saidnetworks be;-

ing so proportioned that the quantity V L [mo-Mora sci-flan 1 has all of, its minimum values equal. 5. A transmission. network comprising two paths connected in parallel between a pair of input terminals and a pair of output terminals;

two frequency selective networks h'a'vingsubstantially, complementary transmission bands dis- F posed in tandem in'on'e oi-"sai-d'paths and two respectively similar networksidisposed in tandem in reverse order in the other of said paths, said,

similar pairs being 'eacrr characterized by two other similar pair beinglikewise characterized by reactances Xe and Xe, the magnitude of said ,networks' being composedof substantially pure: 'reactance' elements, the networks of one of the 7'0 reactances. Xa and Xb'eachhaving a plurality of critical frequencies, and the networks of the reactances and the values of their critical frequencies being so proportioned that the quantity 1 i) i i b e a has all of its minimum values equal.

6. A transmission network comprising two paths connected in parallel between a pair of input terminals and a pair of output terminals, two frequency selective networks having substantially complementary transmission bands disposed in tandem in one of said paths, and two respectively similar networks disposed in tandem in reverse order in the other of said paths, said networks being composed of substantially pure reactance elements, the' networks of one of the similar pairs being each characterized by two reactances Xa and Xb each having a plurality of critical frequencies, and the networks of the other similar pair being likewise characterized by reactances X0 and X1, the magnitude of said reactances and the values of their critical frequencies being so proportioned that the quantity has all of its minimum values equal.

7. A transmission network comprising two paths connected in parallel between a pair of input terminals and a pair of output terminals, two frequency selective networks having substantially complementary transmission bands disposed in tandem in one of said paths, andtwo'respectively similar networks disposed in tandem in reverse order in the other of said paths, said networks being composed of substantially pure reactance elements, the networks of one of the similar pairs being each characterized by two reactances Xa, and Xb each having a plurality of critical frequencies, and the networks of the other similar pair being likewise characterized by reactances X0 and Xd, the magnitude of said reactances and the values of their critical frequencies being so proportioned that the quantity b has a plurality of infinite values alternating with uniform minimum values in a prescribed frequency range, and the quantity i i X: d has a like plurality of infinite values at frequencies in an adjacent range and has minimum values equal to those of the first mentioned quantity.

8. A transmission network comprising two paths connected in parallel between a pair of input terminals and a pair of output terminals,

two frequency selective networks having substan-- tially complementary transmission bands disposed in tandem in one of said paths, and two respectively similar networks disposed in tandem in reverse order in the other of said paths, said networks being composed of substantially pure reactance elements, the networks of one of the similar pairs being each characterized by two reactances Xa and Xb each having a plurality of critical frequencies, and the networks of the other similar pair being likewise characterized by reactances Xc and Xd, the magnitude of said,

reactances and the values of their critical frequencies being so proportioned that the quantity (Xa-Xb) has a plurality of infinite values alternating with uniform minimum values in a prescribed frequency range, and the quantity (XcXd) 9. A transmission network comprising two paths connected in parallel between a pair of input terminals and a pair of output terminals, two frequency selective networks having substantially complementary transmission bands disposed inrtandem in one of said paths, and two respectively similar networks disposed in tandem in reverse order in the other of said paths, said networks being composed of substantially pure reactance'elements, the networksof one of the similar pairs being each characterized by two reactances Xe and X1) each having a plurality of critical frequencies, and the networks of the other similar pair being likewise characterized by reactances X0 and X the said reactances X9. and Xb having poles alternately at frequencies in a prescribed range having the values 10. A transmission network comprising two paths connected in parallel between a pairof input terminals and a pair of output terminals, two frequency selective networks having substantially complementary transmission bands disposed in tandem in one of said paths, and two respectively similar networks disposed in tandem in reverse order in the other of said paths, said networks being composed of substantially pure reactance elements, the networks of one of the similar'pairs being each characterized by two reactances Xe and Xt, each having a plurality of critical frequencies, and the networks of the other similar pair being likewise characterized by reactances X0 and Xd, the susceptances X, and

having poles alternately at frequencies in a prescribed range having the values wherein f0 is a frequency marking one end of the prescribed range, m is the total number of poles, and s is an integer taking the successive values 1 to m, and the susceptances and have a like number of poles alternately atfre quencies in an adjacent range having values intwo frequency s'elective'netwo'rks having substanversely related by a common constant .to those susceptances V and f11.'A transmission network comprising two paths extending in parallel between a pair of input terminalsand a pair of output terminals,

tially complementary transmission bands disposed in'tandem mom of said paths, and two respecbeing likewise tively similar networks disposed in tandem in reworks of one of thejsimilar pairsbeing defined verse order in the other of said paths, the netby two frequency"characteristics 1(w)f and f2(w) which jointly determine their transmission prop- 7 networks of the other similar pair defined by frequency characteristics f3(w)' and ,f4(w)', and the said networks being ertis, and the 7 so proportioned that the poles of the quantity 25 7 at frequencies having substantially the values V 3 5'; V a

' has a like nnmberlof poles in an liewithin a prescribed frequency'rangeand occur l 2s V ra /E314 3, k) wherein fo is a frequency marking one end ofthe prescribed range, mf'is the total number of poles, and s is an" integer takinglthe' successive values l tom, and the quantity 7 7 V r V occurring'at frequencies inversely related by a l f common constant toethose of the first mentioned quantity. 54

I '12. A constan resistance network combination comprising a pairof symmetrical frequency selective networks having substantially complelm'entary transmission bands, one

pair or terminals of each of said networksbeing connected-to 7 i a common pair of inputiterminals, equal resist: 1

the otherpairs of terminals and additional terances connected to of said network's respectively,

minating impedances connected ineparallel with said resistancesthe' additional impedance connected' to each network being equal circuitimpedance of the other network. 7

13. A constant resistance system in accordance with claim 12 in which the said frequencyselctive' networks are proportioned in accordance" with the relation V l 1 1" z @ZZ iZfiYm where R' is the' valuel of the resistances connected 7 to the network terminals, Za and Z5 are respebtively the open circuit andshort-"circuit'impedances of one networkandYo and 'Ys'are respectively the'open circuit and short cir'c'uitimpedances of the other network. V

'14. A wave transmission network having six terminals arranged in ,three pairs, comprising to the Jshort- 7 three transmission paths arranged to intercon- V nect each par of terminalswith eachother pair;

frequency selective networks having substantially complementary transmissionbands included rel-,5 spectively in two of said paths and a network comprising two responding tosa'id complementary networks -in-,j eluded infth'e third of said paths,'said comple tandem connected portions cormentary networks beingof symmetrical struc short-circuit imp edances network and Y0 and Y5 impedances of the other complementary network.

SIDNEY DAaLINGroN e V V V arbitrarily assigned resistance;

Z0 and 25' are respectively the open circuit and of one complementary are the corresponding 

